Affine Hypersurfaces with Parallel Shape Operator

Affine Hypersurfaces with Parallel Shape Operator We prove the following: a relative hypersurface with parallel shape operator is either a relative hypersphere, or it is affinely equivalent to an example constructed by Th. Binder. Furthermore, based on Binder’s example, we give another simple and more explicit example; this way we improve the classification and show that it is completely determined by functions $$\kappa (t)$$ κ ( t ) and $$C(v_i;t)$$ C ( v i ; t ) , the latter being solutions of certain Monge–Ampère equations. Our example geometrically is constructed from a plane curve and a family of relative hyperspheres. In case of an affine sphere with Blaschke geometry we show that our classification can be considered as a construction coming from a plane curve together with a family of improper affine hyperspheres. Especially in $$\mathbb R^4$$ R 4 , this construction is determined by only three functions of a single variable. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Results in Mathematics Springer Journals

Affine Hypersurfaces with Parallel Shape Operator

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Publisher
Springer International Publishing
Copyright
Copyright © 2016 by Springer International Publishing
Subject
Mathematics; Mathematics, general
ISSN
1422-6383
eISSN
1420-9012
D.O.I.
10.1007/s00025-016-0637-5
Publisher site
See Article on Publisher Site

Abstract

We prove the following: a relative hypersurface with parallel shape operator is either a relative hypersphere, or it is affinely equivalent to an example constructed by Th. Binder. Furthermore, based on Binder’s example, we give another simple and more explicit example; this way we improve the classification and show that it is completely determined by functions $$\kappa (t)$$ κ ( t ) and $$C(v_i;t)$$ C ( v i ; t ) , the latter being solutions of certain Monge–Ampère equations. Our example geometrically is constructed from a plane curve and a family of relative hyperspheres. In case of an affine sphere with Blaschke geometry we show that our classification can be considered as a construction coming from a plane curve together with a family of improper affine hyperspheres. Especially in $$\mathbb R^4$$ R 4 , this construction is determined by only three functions of a single variable.

Journal

Results in MathematicsSpringer Journals

Published: Dec 23, 2016

References

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